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A priori error estimates of a Jacobi spectral method for nonlinear systems of fractional boundary value problems and related Volterra-Fredholm integral equations with smooth solutions

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Abstract

Our aim in this paper is to develop a Legendre-Jacobi collocation approach for a nonlinear system of two-point boundary value problems with derivative orders at most two on the interval (0,T). The scheme is constructed based on the reduction of the system considered to its equivalent system of Volterra-Fredholm integral equations. The spectral rate of convergence for the proposed method is established in both L2- and \( L^{\infty } \)- norms. The resulting spectral method is capable of achieving spectral accuracy for problems with smooth solutions and a reasonable order of convergence for non-smooth solutions. Moreover, the scheme is easy to implement numerically. The applicability of the method is demonstrated on a variety of problems of varying complexity. To the best of our knowledge, the spectral solution of such a nonlinear system of fractional differential equations and its associated nonlinear system of Volterra-Fredholm integral equations has not yet been studied in literature in detail. This gap in the literature is filled by the present paper.

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Acknowledgments

The authors wish to thank the referees for their constructive comments and suggestions, which greatly improved the quality of this paper.

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Appendix: Existence of the approximate solution

Appendix: Existence of the approximate solution

We consider the following iteration process:

$$ \begin{array}{llll} \mathbf{U}_{N}^{m}(x) = & \frac{T^{\lambda}}{4^{\lambda} {\Gamma}(\lambda)} \mathcal{I}_{x,N} \left[(x + 1)^{\lambda} {\int}_{-1}^{1} (1 - \eta)^{\lambda-1} \mathcal{I}_{\eta,N}^{\lambda-1,0} \mathbf{G}(\sigma(x,\eta),\mathbf{U}_{N}^{m-1}(\sigma(x,\eta)))d\eta\right]\\ & - \frac{T^{\lambda} (x+1) }{2^{\lambda+1} {\Gamma}(\lambda)} \left[{\int}_{-1}^{1} (1-\beta)^{\lambda-1} \mathcal{I}_{\beta,N}^{\lambda-1,0} \mathbf{G}(\beta,\mathbf{U}_{N}^{m-1}(\beta))d\beta\right]. \end{array} $$
(A.1)

Let \( \overrightarrow {\mathbf {U}}_{N}^{m}(x) = \mathbf {U}_{N}^{m}(x)-\mathbf {U}_{N}^{m-1}(x) \). Then, we have from (A.1) and (4.24) that

$$ \begin{array}{llll} \overrightarrow{\mathbf{U}}_{N}^{m}(x)=& \frac{T^{\lambda}}{2^{\lambda} {\Gamma}(\lambda)} \mathcal{I}_{x,N} \left[ {\int}_{-1}^{x} (x-\sigma)^{\lambda-1} {~}_{x}\widetilde{\mathcal{I}}_{\sigma,N}^{\lambda-1,0} \left( \mathbf{G}\left( \sigma,\mathbf{U}_{N}^{m-1}(\sigma) \right) -\mathbf{G}\left( \sigma,\mathbf{U}_{N}^{m-2}(\sigma) \right) \right) d\sigma \right]\\ &- \frac{T^{\lambda}(x+1)}{2^{\lambda+1} {\Gamma}(\lambda)} {\int}_{-1}^{1} (1-\beta)^{\lambda-1} \mathcal{I}_{\beta,N}^{\lambda-1,0} \left( \mathbf{G}\left( \beta,\mathbf{U}_{N}^{m-1}(\beta) \right) -\mathbf{G}\left( \beta,\mathbf{U}_{N}^{m-2}(\beta) \right) \right) d\beta \\ &=:B_{1} +B_{2}, \end{array} $$
(A.2)

where

$$ \begin{array}{llll} & \mathbf{B}_{1}= \frac{T^{\lambda}}{2^{\lambda} {\Gamma}(\lambda)} \mathcal{I}_{x,N} \left[ {\int}_{-1}^{x} (x-\sigma)^{\lambda-1} {~}_{x}\widetilde{\mathcal{I}}_{\sigma,N}^{\lambda-1,0} \left( \mathbf{G}\left( \sigma,\mathbf{U}_{N}^{m-1}(\sigma) \right) -\mathbf{G}\left( \sigma,\mathbf{U}_{N}^{m-2}(\sigma) \right) \right) d\sigma \right],\\ & \mathbf{B}_{2}=\frac{T^{\lambda}(x+1)}{2^{\lambda+1} {\Gamma}(\lambda)} {\int}_{-1}^{1} (1-\beta)^{\lambda-1} \mathcal{I}_{\beta,N}^{\lambda-1,0} \left( \mathbf{G}\left( \beta,\mathbf{U}_{N}^{m-1}(\beta) \right) -\mathbf{G}\left( \beta,\mathbf{U}_{N}^{m-2}(\beta) \right) \right) d\beta. \end{array} $$

We obtain from the Cauchy-Schwarz inequality that

$$ \begin{array}{llll} &\left\|\mathbf{B}_{1} \right\| \\ &=\frac{T^{\lambda}}{2^{\lambda} {\Gamma}(\lambda)} \left\| \mathcal{I}_{x,N} {\int}_{-1}^{x} (x-\sigma)^{\lambda-1} {~}_{x}\widetilde{\mathcal{I}}_{\sigma,N}^{\lambda-1,0} \left( \mathbf{G}\left( \sigma,\mathbf{U}_{N}^{m-1}(\sigma) \right) -\mathbf{G}\left( \sigma,\mathbf{U}_{N}^{m-2}(\sigma) \right) \right) d\sigma \right\|\\ &= \frac{T^{\lambda}}{2^{\lambda} {\Gamma}(\lambda)} \left( {\int}_{-1}^{1} \left( \sum\limits_{q=1}^{Q} \mathcal{I}_{x,N} {\int}_{-1}^{x} (x-\sigma)^{\lambda-1} {~}_{x}\widetilde{\mathcal{I}}_{\sigma,N}^{\lambda-1,0} \left( G_{q}\left( \sigma,\mathbf{U}_{N}^{m-1}(\sigma)\right)-G_{q}\left( \sigma,\mathbf{U}_{N}^{m-2}(\sigma)\right) \right)d\sigma \right)^{2} dx\right)^{\frac{1}{2}}\\ &=\frac{T^{\lambda}}{2^{\lambda} {\Gamma}(\lambda)}\left( \sum\limits_{j=0}^{N} \varpi_{j} \left( {\int}_{-1}^{x_{j}} (x_{j}-\sigma)^{\lambda-1}\sum\limits_{q=1}^{Q} {~}_{x_{j}}\widetilde{\mathcal{I}}_{\sigma,N}^{\lambda-1,0} \left( G_{q}\left( \sigma, \mathbf{U}_{N}^{m-1}(\sigma) \right)-G_{q}\left( \sigma,\mathbf{U}_{N}^{m-2}(\sigma)\right) \right)d\sigma \right)^{2}\right)^{\frac{1}{2}} \\ & \leq \frac{T^{\lambda}}{2^{\lambda} {\Gamma}(\lambda)} \left( \sum\limits_{j=0}^{N} \varpi_{j} {\int}_{-1}^{x_{j}} (x_{j}-\sigma)^{\lambda-1} d\sigma {\int}_{-1}^{x_{j}}(x_{j}-\sigma)^{\lambda-1} \right.\\ & \left. \qquad \qquad \qquad \times \left( \left| \sum\limits_{q=1}^{Q} {~}_{x_{j}}\widetilde{\mathcal{I}}_{\sigma,N}^{\lambda-1,0} \left( G_{q}(\sigma, \mathbf{U}_{N}^{m-1}(\sigma) )-G_{q}(\sigma,\mathbf{U}_{N}^{m-2}(\sigma))\right) \right| \right)^{2} d\sigma\right)^{\frac{1}{2}} \\ & \leq \frac{T^{\lambda}}{2^{\lambda} {\Gamma}(\lambda)} \left( \sum\limits_{j=0}^{N} \varpi_{j} \frac{(x_{j}+1)^{\lambda}}{\lambda} {\int}_{-1}^{x_{j}}(x_{j}-\sigma)^{\lambda-1}\right.\\ & \left. \qquad \qquad \qquad \times \left( \sum\limits_{q=1}^{Q} \left| {~}_{x_{j}}\widetilde{\mathcal{I}}_{\sigma,N}^{\lambda-1,0} \left( G_{q}(\sigma, \mathbf{U}_{N}^{m-1}(\sigma) )-G_{q}(\sigma,\mathbf{U}_{N}^{m-2}(\sigma))\right) \right| \right)^{2} d\sigma \right)^{\frac{1}{2}} \\ & \leq \frac{T^{\lambda}}{2^{\lambda} {\Gamma}(\lambda)} \left( \sum\limits_{j=0}^{N} \varpi_{j} \frac{(x_{j}+1)^{2\lambda}}{2^{\lambda}\lambda} \right.\\ & \left. \qquad \qquad \qquad \times \sum\limits_{k=0}^{N} \left( \sum\limits_{q=1}^{Q} \left|G_{q}(\sigma_{k}^{\lambda-1,0},\mathbf{U}_{N}^{m-1}(\sigma_{k}^{\lambda-1,0}))-G_{q}(\sigma_{k}^{\lambda-1,0},\mathbf{U}_{N}^{m-2}(\sigma_{k}^{\lambda-1,0})) \right|\right)^{2} \varpi_{k}^{\lambda-1,0}\right)^{\frac{1}{2}}. \end{array} $$

Hence, by (4.25), (5.14) and the Lipschitz condition, we obtain that

$$ \begin{array}{llll} & \left\|\mathbf{B}_{1} \right\|\\ & \leq \frac{T^{\lambda}}{2^{\lambda} {\Gamma}(\lambda)} \left( \sum\limits_{j=0}^{N} \varpi_{j} \frac{(x_{j}+1)^{2\lambda}}{2^{\lambda}\lambda} \sum\limits_{k=0}^{N} \left( \sum\limits_{q=1}^{Q} \sum\limits_{i=1}^{Q} L_{q,i} \left| U_{N,i}^{m-1}(\sigma_{k}^{\lambda-1,0})-U_{N,i}{m-2}(\sigma_{k}^{\lambda-1,0}) \right|\right)^{2} \varpi_{k}^{\lambda-1,0} \right)^{\frac{1}{2}}\\ & = \frac{\lambda}{ 2^{\lambda+1}}\left( \sum\limits_{j=0}^{N} \varpi_{j} \frac{(x_{j}+1)^{2\lambda}}{2^{\lambda}\lambda} \sum\limits_{k=0}^{N} \left( \sum\limits_{i=1}^{Q} \left| U_{N,i}^{m-1}(\sigma_{k}^{\lambda-1,0})-U_{N,i}^{m-2}(\sigma_{k}^{\lambda-1,0}) \right|\right)^{2} \varpi_{k}^{\lambda-1,0} \right)^{\frac{1}{2}}\\ & \leq \frac{\lambda}{ 2^{\lambda+1}} \left( \sum\limits_{j=0}^{N} \varpi_{j} \frac{(x_{j}+1)^{\lambda}}{\lambda} {\int}_{-1}^{x_{j}} (x_{j}-\sigma)^{\lambda-1} \left( \sum\limits_{i=1}^{Q}\left| {~}_{x_{j}}\widetilde{\mathcal{I}}_{\sigma,N}^{\lambda-1,0} \overrightarrow{U}_{N,i}^{m-1}(\sigma) \right|\right)^{2} d\sigma\right)^{\frac{1}{2}}\\ & \leq \frac{\lambda}{ 2^{\lambda+1}} \left( \sum\limits_{j=0}^{N} \varpi_{j} \frac{(x_{j}+1)^{\lambda}}{\lambda}\right)^{\frac{1}{2}} \max\limits_{0\leq j \leq N} \left( {\int}_{-1}^{x_{j}} (x_{j}-\sigma)^{\lambda-1} \left( \sum\limits_{i=1}^{Q} \left| \overrightarrow{U}_{N,i}^{m-1}(\sigma) \right|\right)^{2} d\sigma\right)^{\frac{1}{2}}\\ & \leq \frac{\lambda}{ 2^{\lambda+1}} \sqrt{\frac{8}{3 \lambda}} \max\limits_{0\leq j \leq N} \left( {\int}_{-1}^{x_{j}} (x_{j}-\sigma)^{\lambda-1} \left( \sum\limits_{i=1}^{Q} \left| \overrightarrow{U}_{N,i}^{m-1}(\sigma) \right|\right)^{2} d\sigma\right)^{\frac{1}{2}}\\ &\leq \frac{\lambda}{ 2^{\lambda+1}} \sqrt{\frac{2^{\lambda+2}}{3 \lambda}} \left( {\int}_{-1}^{1} \left( \sum\limits_{i=1}^{Q} \left| \overrightarrow{U}_{N,i}^{m-1}(\sigma) \right| \right)^{2} d\sigma\right)^{\frac{1}{2}}\\ &\leq \sqrt{\frac{\lambda}{3 \times 2^{\lambda} }} \left\| \overrightarrow{\mathbf{U}}^{m-1} \right\|. \end{array} $$
(A.3)

It remains to estimate the term \( \left \|\boldsymbol {B}_{2} \right \| \). By the Cauchy-Schwarz inequality, we have

$$ \begin{array}{llll} \left\|\boldsymbol{B}_{2} \right\| &= \frac{T^{\lambda} \left\|x+1\right\|}{2^{\lambda+1} {\Gamma}(\lambda)} \left|{\int}_{-1}^{1} (1-\beta)^{\lambda-1} \mathcal{I}_{\beta,N}^{\lambda-1,0} \left( \mathbf{G}\left( \beta,\mathbf{U}_{N}^{m-1}(\beta) \right) -\mathbf{G}\left( \beta,\mathbf{U}_{N}^{m-2}(\beta) \right) \right) d\beta\right|\\ &= \frac{T^{\lambda}}{2^{\lambda+1} {\Gamma}(\lambda)} \sqrt{\frac{2^{\lambda+3}}{3 \lambda}} \left|{\int}_{-1}^{1} \sum\limits_{q=1}^{Q} (1-\beta)^{\lambda-1} \mathcal{I}_{\beta,N}^{\lambda-1,0} \left( G_{q}(\beta,\mathbf{U}_{N}^{m-1}(\beta))-G_{q}(\beta,\mathbf{U}^{m-2}(\beta))\right) d\beta\right|\\ & \leq \frac{T^{\lambda}}{2^{\lambda+1} {\Gamma}(\lambda)} \sqrt{\frac{2^{\lambda+3}}{3 \lambda}} \left( {\int}_{-1}^{1} (1-\beta)^{\lambda-1} \left( \sum\limits_{q=1}^{Q} \left| \mathcal{I}_{\beta,N}^{\lambda-1,0} \left( G_{q}(\beta,\mathbf{U}^{m-1}_{N}(\beta))-G_{q}(\beta,\mathbf{U}^{m-2}(\beta))\right) d\beta\right| \right)^{2} \right)^{\frac{1}{2}} \end{array} $$

The previous result, along with (2.8) and Lipschitz condition, yields

$$ \begin{array}{llll} \left\|\boldsymbol{B}_{2} \right\| & \leq \frac{T^{\lambda}}{2^{\lambda+1} {\Gamma}(\lambda)} \sqrt{\frac{2^{\lambda+3}}{3 \lambda}} \left( \sum\limits_{j=0}^{N} \varpi_{j}^{\lambda-1 ,0 } \left( \sum\limits_{q=1}^{Q} \left| \left( G_{q}(x_{j}^{\lambda-1,0},\mathbf{U}_{N}^{m-1}(x_{j}^{\lambda-1,0}))-G_{q}(x_{j}^{\lambda-1,0},\mathbf{U}_{N}^{m-2}(x_{j}^{\lambda-1,0}))\right) \right|\right)^{2}\right)^{\frac{1}{2}}\\ & \leq \frac{T^{\lambda}}{2^{\lambda+1} {\Gamma}(\lambda)} \sqrt{\frac{2^{\lambda+3}}{3 \lambda}} \left( \sum\limits_{j=0}^{N} \varpi_{j}^{\lambda-1 ,0 } \left( \sum\limits_{q=1}^{Q} \sum\limits_{i=1}^{Q} L_{q,i} \left| U_{N,i}^{m-1}(x_{j}^{\lambda-1,0}))-U_{N,i}^{m-2}(x_{j}^{\lambda-1,0})) \right|\right)^{2}\right)^{\frac{1}{2}}\\ & \leq \frac{ \lambda }{2^{\lambda+2}} \sqrt{\frac{2^{\lambda+3}}{3 \lambda}} \left( \sum\limits_{j=0}^{N} \varpi_{j}^{\lambda-1 ,0 } \left( \sum\limits_{i=1}^{Q} \left| \overrightarrow{U}_{N,i}^{m-1}(x_{j}^{\lambda-1,0}) \right|\right)^{2}\right)^{\frac{1}{2}}\\ & = \frac{ \lambda }{2^{\lambda+2}} \sqrt{\frac{2^{\lambda+3}}{3 \lambda}} \left( {\int}_{-1}^{1} (1-\beta)^{\lambda-1}\left( \sum\limits_{i=1}^{Q} \left| \overrightarrow{U}_{N,i}^{m-1}(\beta)\right|\right)^{2} d\beta\right)^{\frac{1}{2}} \\ & \leq \sqrt{\frac{\lambda}{12}} \left\| \overrightarrow{\mathbf{U}}^{m-1} \right\|, \qquad \qquad \forall \lambda \in (1,2). \end{array} $$
(A.4)

Hence,

$$ \left\| \overrightarrow{\mathbf{U}}_{N}^{m} \right\|\le \left( \sqrt{\frac{\lambda}{3 \times 2^{\lambda} }}+\sqrt{\frac{\lambda}{12}} \right) \left\| \overrightarrow{\mathbf{U}}^{m-1} \right\|. $$

since

$$ \sqrt{\frac{\lambda}{3 \times 2^{\lambda} }}+\sqrt{\frac{\lambda}{12}} < 1, \qquad \qquad \forall \lambda \in (1,2), $$

we have \( \left \| \overrightarrow {\mathbf {U}}_{N}^{m} \right \| \longrightarrow 0 \) as \( m\longrightarrow \infty . \) It implies the existence of solution of (3.10).

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Zaky, M.A., Ameen, I.G. A priori error estimates of a Jacobi spectral method for nonlinear systems of fractional boundary value problems and related Volterra-Fredholm integral equations with smooth solutions. Numer Algor 84, 63–89 (2020). https://doi.org/10.1007/s11075-019-00743-5

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